DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 403 



Canonical Form of Differential Etptation and of the Invariants. 



29. The conclusion as to the general form of the invariants suggests that a form of 

 the differential equation might be adopted which would give to tlie invariants an 

 expression considerably simplified. If it were possible to take Q e to be zero (and this 

 possibility will be proved immediately), then, when the function 6, (z) is formed from 

 the coefficients of the transformed equation (ii.), the non-linear part of the function 

 will cease to appear because every term in this part contains a vanishing factor ; and 

 the part that remains constitutes the whole of the invariant. Hence, for the differential 

 equation, thus transformed, the invariant is a purely linear function of its coefficients, 

 and, in this linear form, the invariant is determinate wheu once the numerical coeffi- 

 cients are obtained. 



30. In order to obtain the transformed differential equation, the invariants for 

 which have this simple form, we return to the original equations of transformation. 

 By them, considered as applied to equation (i.), there were two quantities at our 

 disposal, viz., X and 2. One relation between them has already been assumed in 

 deducing the equation (5) or (iv.) ; and any other may be taken to completely deter- 

 mine them, provided it does not violate that already adopted. Such a relation which 

 is permissible is to suppose the quantities X and z, already subject to (iv.), to be quan- 

 tities which will make the coefficient of ^""^/t/z"" 2 in (ii.) zero, that is, make Q 3 = 0. 

 Hence, by (6), we must have 



"""'' : '"V"" 1 ' 2Z ' = z ' + ;rTT p " "' ' : " : 



where, by (8), 



*" 7 2 V 



7'- "^rix' 



If we write 



the equation which determines Z is transformed into 



and then we may write 



X=^- 1 , z'=0- 2 ........ (20). 



Hence, by the solution of a linear differential equation of the second order, as (19), 

 the two terms of orders next below the highest can be removed from a linear differeu- 

 tial equation of any order.* This modified form may be called the canonical form of 

 the differential equation. 



* This result has already been referred to, as a general statement ( 2 ) ; the exact references ore 

 COCKLE, ' Quarterly Journal of Mathematics,' vol. 14, 1876, p. 34C, for the cubic; 

 LAGUKBRK, ' CompHs Rendus,' vol. 88, 1879, p. 226, for the general equation. 



3 F 2 



